Higher. Further Calculus 149

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1 hsn.uk.net Higher Mathematics UNIT 3 OUTCOME 2 Further Calculus Contents Further Calculus 49 Differentiating sinx an cosx 49 2 Integrating sinx an cosx 50 3 The Chain Rule 5 4 Special Cases of the Chain Rule 5 5 A Special Integral 54 Integrating sin(ax + b) an cos(ax + b) 57 This ocument was prouce specially for the HSN.uk.net website, an we require that any copies or erivative works attribute the work to Higher Still Notes. For more etails about the copyright on these notes, please see

2 OUTCOME 2 Further Calculus Differentiating sinx an cosx In orer to ifferentiate expressions involving trigonometric functions, we use the following rules: ( sin x ) = cos x, ( cos x ) = sin x. These rules only work when x is an angle measure in raians. A form of these rules is given in the exam.. Differentiate y = 3sin x with respect to x. 2. A function f is efine by f ( x ) = sin x 2cos x for x R. Fin f π Fin the equation of the tangent to the curve y = sin x when x = π. Page 49

3 2 Integrating sinx an cosx We know the erivatives of sin x an cos x, so it follows that the integrals are: cos x = sin x + c, sin x = cos x + c. Again, these results only hol if x is measure in raians.. Fin ( 5sin x + 2cos x ). x Fin ( 4cos x + 2sin ) 0 π 3. Fin the value of sin x. Remember We must use raians when integrating or ifferentiating trig. functions. Page 50

4 3 The Chain Rule We will now look at how to ifferentiate composite functions, such as f g ( x ). If the functions f an g are efine on suitable omains, then f ( g ( x )) = f g ( x ) g x. State simply: ifferentiate the outer function, the bracket stays the same, then multiply by the erivative of the bracket. This is calle the chain rule. You will nee to remember it for the exam. EXAMPLE If y cos( 5x π ) = +, fin y. 4 Special Cases of the Chain Rule We will now look at how the chain rule can be applie to particular types of expression. Powers of a Function [ ] n For expressions of the form f ( x ), where n is a constant, we can use a simpler version of the chain rule: [ ] n n ( f ( x )) n f ( x ) f = x State simply: the power ( n ) multiplies to the front, the bracket stays the same, the power lowers by one (giving n ) an everything is multiplie by the erivative of the bracket ( f ( x )).. Page 5

5 2. A function f is efine on a suitable omain by f ( x ) = 2x + 3x. Fin f ( x ). 2. Differentiate y 4 = 2sin x with respect to x. Powers of a Linear Function The rule for ifferentiating an expression of the form ( ax + b, where a, b an n are constants, is as follows: n n ax b an ax b. ( + ) = ( + ) 3. Differentiate y = ( 5x + 2) 3 with respect to x. ) n Page 52

6 4. If y = ( 2x + ) 3, fin y. 5. A function f is efine by f ( x ) = 3 ( 3x 2) 4 for x R. Fin f ( x ). Trigonometric Functions The following rules can be use to ifferentiate trigonometric functions. sin( ax + b) = a cos( ax + b) cos( ax + b) = a sin( ax + b) These are given in the exam. EXAMPLE. Differentiate y = sin( 9x + π) with respect to x. Page 53

7 5 A Special Integral The metho for integrating an expression of the form ( ax + b is: ( + ) n+ n ax b ( ax + b) = + c where a 0 an n. a( n + ) State simply: raise the power ( n ) by one, ivie by the new power an also ivie by the erivative of the bracket ( a ( n + )), a c. x Fin 7 ) n 2. Fin ( 2x + 3 ) Fin x + where x 9 5. Page 54

8 4. Evaluate 3 3x + 4 where x Warning Make sure you on t confuse ifferentiation an integration this coul lose you a lot of marks in the exam. Remember the following rules for ifferentiating an integrating expressions of the form ( ax + b) n : n n ax b an ax b, ( + ) = ( + ) ( + ) n+ n ax b ax + b = + c. a( n + ) These rules will not be given in the exam. Page 55

9 Using Differentiation to Integrate Recall that integration is just the process of unoing ifferentiation. So if we ifferentiate f ( x ) to get g ( x ) then we know that g ( x ) = f ( x ) + c. 5. (a) Differentiate 5 y = ( 3x ) 4 (b) Hence, or otherwise, fin with respect to x. ( 3x ) 5.. (a) Differentiate (b) Hence, fin y = 3 ( x ) 5 x 2 3 ( x ). with respect to x. Page 5

10 Integrating sin(ax + b) an cos(ax + b) Since we know the erivatives of sin( ax + b) an cos( ax b) that their integrals are: These are given in the exam.. Fin sin( 4x + ). cos ax + b = a sin ax + b + c, sin ax + b = a cos ax + b + c. +, it follows 2. Fin cos 3 ( x + π ) Fin the value of cos ( 2x 5 ). 0 Remember We must use raians when integrating or ifferentiating trig. functions. Page 57

11 4. Fin the area enclose by the graph of y sin( 3x π ) the lines x = 0 an x = π. y = +, the x-axis an y = sin 3x + π O π x Fin 2cos( x 3 ) x x.. Fin 5cos( 2 ) + sin( 3 ) Page 58

12 7. (a) Differentiate (b) Hence fin cos x tan x. cos x with respect to x. Page 59